Notes

If showers were physically painful, people would obviously not take them as opposed to baths (assuming baths aren't just as painful).
Therefore, the purpose of showers is to be pleasureable? No.
If using a marker to write was physically painful, people would obviously use pens and pencils instead (assuming pens aren't just as painful).
Therefore, the purpose of markers is to be pleasureable? No.

The purpose of showers is not to be pleasureable, however, that doesn' t mean showers can't be warm and relaxing. The first showers ever in history were, in fact, cold showers. The first showers ever taken were in waterfalls, which are cold. The first manmade showers ever taken (by the Greeks) were cold, and likely unpleasurable. The appeal of showers over baths was that they were more efficient and easier to execute. The first baths did not have drainage way back when, and there were no spigots. Both the filling and removal of bath water had to be done manually, and water is very heavy. Therefore, people started pouring water over themselves to make bathing easier. They were purely a matter of convinience.

Also, with the advent of water heaters, baths would STILL probably be more fun than showers. That is because when you are taking a shower, there is a lot of air touching you, which makes you cold--evaporation is a cooling process. In baths, it's all hot water around you which feels nice. Would you rather stand under hot water dripping on you, or sit in a nice, warm hot tub? Probably the hot tub.

Therefore, taking ice cold showers doesn't somehow defeat the purpose of showers. Showers can be nice and warm, however, that is not the sole reason for their existence. I rest my case.